Percolate a regular graph: keep each vertex independently with probability p. In the supercritical regime — p above the threshold — a giant component emerges. But how giant, and how unique?
Diskin, Krivelevich, and Markbreit show that under sufficient conditions on d-regular graphs, the Erdős-Rényi component phenomenon appears: exactly one giant component of order n/d, with every other component at most O(log n). The gap between the giant and the second-largest is not a matter of degree — it is exponential. There is no runner-up.
The result applies to d-dimensional hypercubes and pseudo-random graphs, resolving two previously open questions. The hypercube is particularly interesting because its local structure is sparse — every vertex has only d neighbors — yet the global connectivity produces the same sharp giant-component behavior seen in dense random graphs. The regularity of the degree sequence forces a clean phase transition regardless of the specific graph geometry.
The structural insight: uniqueness is not a consequence of the giant component being large. It is a consequence of the percolation threshold being sharp. Below the threshold, all components are logarithmic. Above it, one component captures a constant fraction of the vertices. The transition is too sharp for two components to simultaneously grow to macroscopic size. The system either fragments completely or consolidates around a single cluster — never two.